SOLUTION: Please find derivative and simplfy: y= x^3e^-2x y= square root e^x y= square root lnx y= ln square root x y= e^x^2e^2x Thanks alot! i'm so confused!

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: Please find derivative and simplfy: y= x^3e^-2x y= square root e^x y= square root lnx y= ln square root x y= e^x^2e^2x Thanks alot! i'm so confused!       Log On


   

Question 59400: Please find derivative and simplfy:
y= x^3e^-2x
y= square root e^x
y= square root lnx
y= ln square root x
y= e^x^2e^2x
Thanks alot! i'm so confused!
Answer by funmath(2933) About Me  (Show Source):
You can put this solution on YOUR website!
d/dx[x^n]=nx^(n-1)
d/dx[cf(x)]=cf'(x)
d/dx[ln(x)]=1/x
d/dx[e^x]=e^x
Product rule: d/dx[f(x)g(x)]=f(x)g'(x)+g(x)f'(x)
Chain Rule: d/dx[f(g(x))]=f'(g(x))*g'(x)
Please find derivative and simplfy:
:
y=x%5E3e%5E-2x
dy%2Fdx=x%5E3%28dy%2Fdx%29%28e%5E%28-2x%29%29%2Be%5E%28-2x%29%28dy%2Fdx%29%28x%5E3%29 Product rule
dy%2Fdx=x%5E3%2Ae%5E%28-2x%29%28dy%2Fdx%29%28-2x%29%2Be%5E%28-2x%29%2A%283x%5E2%29 Chain rule
dy%2Fdx=x%5E3%2Ae%5E%28-2x%29%28-2%29%2Be%5E%28-2x%29%2A%283x%5E2%29
dy%2Fdx=-2x%5E3%2Ae%5E%28-2x%29%2B3x%5E2%2Ae%5E%28-2x%29 Many Teachers are happy now and you can stop.
highlight%28dy%2Fdx=x%5E2%2Ae%5E%28-2x%29%2A%283-2x%29%29 A lot of books want you to factor it like this.
:
y=sqrt%28e%5Ex%29---> y=%28e%5Ex%29%5E%281%2F2%29 ---> y=e%5E%28%281%2F2%29x%29
dy%2Fdx=e%5E%28%281%2F2%29x%29%2A%28dy%2Fdx%29%281%2F2%29x Chain Rule
dy%2Fdx=e%5E%28%281%2F2%29x%29%2A%281%2F2%29
dy%2Fdx=%281%2F2%29%2Ae%5E%28%281%2F2%29x%29 Usually you can stop here.
highlight%28dy%2Fdx=%281%2F2%29%2Asqrt%28e%5Ex%29%29
:
y=sqrt%28ln%28x%29%29-->y=%28ln%28x%29%29%5E%281%2F2%29
dy%2Fdx=%281%2F2%29%28ln%28x%29%29%5E%281%2F2-1%29%2A%28dy%2Fdx%29%28ln%28x%29%29 Chain Rule
dy%2Fdx=%281%2F2%29%28ln%28x%29%29%5E%281%2F2-1%29%2A%281%2Fx%29
dy%2Fdx=%281%2F2%29%28ln%28x%29%29%5E%28-1%2F2%29%2Fx Sometimes you can stope here
dy%2Fdx=1%2F%282x%28ln%28x%29%29%5E%281%2F2%29%29 Most of the time you can stop here.
highlight%28dy%2Fdx=1%2F%282x%2Asqrt%28ln%28x%29%29%29%29 A lot of books say this.
:
y=ln%28sqrt%28x%29%29--->y=ln%28x%5E%281%2F2%29%29---> y=%281%2F2%29ln%28x%29
dy%2Fdx=%281%2F2%29%2A%281%2Fx%29
dy%2Fdx=1%2F2x
:
y=e%5Ex%5E2e%5E2x --> y=e%5E%28x%5E2%2B2x%29
dy%2Fdx=e%5E%28x%5E2%2B2x%29dy%2Fdx%28x%5E2%2B2x%29 chain rule
dy%2Fdx=e%5E%28x%5E2%2B2x%29%282x%2B2%29 Most teachers are thrilled with this.
highlight%28dy%2Fdx=2e%5E%28x%5E2%2B2x%29%28x%2B1%29%29 You can factor out the 2 if you want to.
:
Happy Calculating!!!